The crystal structure of Rb-III, stable between 13 and 17 GPa, has been determined from quasi-single-crystal x-ray diffraction data. It is orthorhombic, space group C222(1), with 52 atoms in the unit cell, and has an 8-10-8-8-10-8 stacking of 8- and 10-atom layers. The recently reported 84-atom structure of Cs-III can be understood as an 8-8-10-8-8-8-8-10-8-8 stacking of the same layers. These represent a new class of modulated elemental structures.